Perturbation of Closed Range Operators

Let T, A be operators with domains D(T) \subseteq D(A) in a normed space X. The operator A is called T-bounded if |Ax|\leq a|x|+b|Tx| for some a, b\geq 0 and all x \in D(T). If A has the Hyers--Ulam stability then under some suitable assumptions we show that both T and S: = A+T have the Hyers--Ulam stability. We also discuss the best constant of Hyers--Ulam stability for the operator S. Thus we establish a link between T-bounded operators and Hyers--Ulam stability.

Anahtar Kelimeler:

Hilbert space; perturbation; Hyers--Ulam stability; closed operator; semi-Fredholm operator.

Perturbation of Closed Range Operators

Keywords:

Hilbert space; perturbation; Hyers--Ulam stability; closed operator; semi-Fredholm operator.,

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Mohammad Sal MOSLEHIAN1,2, Ghadir SADEGHI1,2 Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box , Mashhad 91775, IRAN e-mail: moslehian@ferdowsi.um.ac.ir and moslehian@ams.org
Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad-IRAN e-mail: ghadir54@yahoo.com